CONVERGENCE_ANALYSIS

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A Convergence Study of SGD-Type Methods for Stochastic Optimization
《Numerical Mathematics(Theory,Methods and Applications)》2023年第4期914-930,共17页Tiannan Xiao Guoguo Yang 
supported by the NSFC(Grant No.11825102);the China Postdoctoral Science Foundation(Grant No.2023M730093);the National Key R&D Program of China(Grant No.2021YFA1003300).
In this paper,we first reinvestigate the convergence of the vanilla SGD method in the sense of L2 under more general learning rates conditions and a more general convex assumption,which relieves the conditions on lear...
关键词：SGD momentum SGD Nesterov acceleration time averaged SGD convergence analysis non-convex 
A Convergence Analysis of a Structure-Preserving Gradient Flow Method for the All-Electron Kohn-Sham Model
《Numerical Mathematics(Theory,Methods and Applications)》2023年第3期597-621,共25页Yedan Shen Ting Wang Jie Zhou Guanghui Hu 
partially funded by the Hunan National Applied Mathematics Center of Hunan Provincial Science and Technology Department(Grant No.2020ZYT003);by the RSF-NSFC Cooperation project(Grant No.12261131501);by the Excellent youth project of the Hunan Education Department(Grant No.19B543);partially supported by the National Natural Science Foundation of China(Grant Nos.11922120 and 11871489);by the FDCT of Macao SAR(Grant No.0082/2020/A2);by the MYRG of the University of Macao(Grant No.MYRG2020-00265-FST);by the Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications(Grant No.2020B1212030001).
In[Dai et al.,Multi.Model.Simul.18(4)(2020)],a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory,based on which a linearized method was deve...
关键词：Kohn-Sham density functional theory gradient flow model structure-preserving linear scheme convergence analysis 
Convergence Analysis of a Quasi-Monte Carlo-Based Deep Learning Algorithm for Solving Partial Differential Equations
《Numerical Mathematics(Theory,Methods and Applications)》2023年第3期668-700,共33页Fengjiang Fu Xiaoqun Wang 
supported by the National Natural Science Foundation of China(Grant No.72071119).
Deep learning has achieved great success in solving partial differential equations(PDEs),where the loss is often defined as an integral.The accuracy and efficiency of these algorithms depend greatly on the quadrature ...
关键词：Deep Ritz method quasi-Monte Carlo Poisson equation static Schrodinger equation error bound 
A Third Order Accurate in Time,BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation
《Numerical Mathematics(Theory,Methods and Applications)》2022年第2期279-303,共25页Kelong Cheng Cheng Wang Steven M.Wise Yanmei Wu 
supported in part by the Computational Physics Key Laboratory of IAPCAM(P.R.China)under Grant 6142A05200103(K.Cheng);the National Science Foundation(USA)under Grant NSF DMS-2012669(C.Wang);Grants NSF DMS-1719854,DMS-2012634(S.Wise).
In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discret...
关键词：Cahn-Hilliard equation third order backward differentiation formula unique solvability energy stability discrete L6 N estimate optimal rate convergence analysis 
On Discontinuous and Continuous Approximations to Second-Kind Volterra Integral Equations
《Numerical Mathematics(Theory,Methods and Applications)》2022年第1期91-124,共34页Hui Liang 
supported by the National Nature Science Foundation of China(No.12171122,11771128);the Fundamental Research Project of Shenzhen(No.JCYJ20190806143201649);Project(HIT.NSRIF.2020056);the Natural Scientific。
Collocation and Galerkin methods in the discontinuous and globally continuous piecewise polynomial spaces,in short,denoted as DC,CC,DG and CG methods respectively,are employed to solve second-kind Volterra integral eq...
关键词：Volterra integral equations collocation methods Galerkin methods discontinuous Galerkin methods convergence analysis 
Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach被引量：1
《Numerical Mathematics(Theory,Methods and Applications)》2020年第1期63-80,共18页Chenghua Duan Chun Liu Cheng Wang Xingye Yue 
The work of Yue is supported in part by NSF of China under the grants No.11971342.
The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],...
关键词：Energetic variational approach porous medium equation trajectory equation optimal rate convergence analysis 
Extremal Eigenvalues of the Sturm-Liouville Problems with Discontinuous Coefficients
《Numerical Mathematics(Theory,Methods and Applications)》2013年第4期657-684,共28页Shuangbing Guo Dingfang Li Hui Feng Xiliang Lu 
supported by the National Natural Science Foundation of China(10971159,91130022,11101316).
In this paper,an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated.Liouville transformation is applied to change the problem into an equ...
关键词：Extremal eigenvalue problem Sturm-Liouville problem finite element method convergence analysis 
A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel被引量：1
《Numerical Mathematics(Theory,Methods and Applications)》2013年第2期424-446,共23页Yunxia Wei Yanping Chen 
supported by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008);National Science Foundation of China(10971074);Specialized Research Fund for the Doctoral Program of Higher Education(20114407110009).
This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel.The solution of this equati...
关键词：Neutral Volterra integro-differential equation weakly singular kernel Jacobi collocation discretization convergence analysis 
Convergence Analysis of a Block-by-Block Method for Fractional Differential Equations被引量：11
《Numerical Mathematics(Theory,Methods and Applications)》2012年第2期229-241,共13页Jianfei Huang Yifa Tang Luis Vázquez 
supported by the State Key Laboratory of Scientific and Engineering Computing,Chinese Academy of Sciences and by Hunan Key Laboratory for Computation and Simulation in Science and Engineering,by National Natural Science Foundation of China(Grant Nos.60931002,11001072 and 11026154);partially by the Spanish Ministry of Science and Innovation under Grant AYA2009-14212-C05-05.
The block-by-block method,proposed by Linz for a kind of Volterra integral equations with nonsingular kernels,and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations...
关键词：Fractional differential equation Caputo derivative block-by-block method convergence analysis 
Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations被引量：3
《Numerical Mathematics(Theory,Methods and Applications)》2011年第3期419-438,共20页Yunxia Wei Yanping Chen 
the Foundation for Talent Introduction of Guangdong Provincial University,Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008),National Science Foundation of China(10971074).
A class of numerical methods is developed for second order Volterra integrodifferential equations by using a Legendre spectral approach.We provide a rigorous error analysis for the proposed methods,which shows that t...
关键词：Second order Volterra integro-differential equations Gauss quadrature formula Legendre-collocation methods convergence analysis. 
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